About the linearization of certain subgroups of polynomial diffeomorphisms of the plane and the envelopes of holomorphy
In the paper the following notations and definitions are used: -- \(\Aut_0[\mathbb{C}^2]\) -- the polynomial automorphisms group of \(\mathbb{C}^2\) with fixed point at the origin, -- a linear automorphism \(A\in \text{GL}(\mathbb{C}^2)\) is said to be: -- contracting if its proper values \(\lambda_1(A)\) and \(\lambda_2(A)\) fullfil the inequalities \(|\lambda_1(A)|< 1\), \(|\lambda_2(A)|< 1\), -- hyperbolic if \(|\lambda_1(A)|< 1< |\lambda_2(A)|\), -- \(\text{Diff}(\mathbb{C}^2, 0)\) -- the set of germs of holomorphic diffeomorphisms at \(0\in\mathbb{C}^2\) and \(\widehat{\text{Diff}}(\mathbb{C}^2, 0)\) -- the formal complement of \(\text{Diff}(\mathbb{C}^2, 0)\), -- a subgroup \(G\) of \(\Aut_0[\mathbb{C}^2]\) is said to be algebraically linearizable if there exist \(\varphi\in\Aut_0[\mathbb{C}^2]\) such that \(F\circ\varphi= \varphi\circ F'(0)\,\forall F\in G\), this subgroup \(G\) is said to be-- locally linearizable if \(\varphi\in \text{Diff}(\mathbb{C}^2, 0)\) -- formally linearizable if \(\varphi\in \widehat{\text{Diff}}(\mathbb{C}^2, 0)\), -- \(j^1: \Aut_0[\mathbb{C}^2]\to \text{GL}(\mathbb{C}^2)\) a morphism assigning to \(F\) ist differential \(F'(0)\) at the origin. The authors discuss the question: which are the subgroups \(G\) that are linearizable? They give several cases of linearization, among other, where \(j^1G\) contains a contracting and a hyperbolic element and the case when \(j^1 G= \text{SL}(2,\mathbb{Z})\).
scientific article; zbMATH DE number 1777201
scientific article